Request for help justifying the use of the word cell instead of ? in our paper that is based on ASPECT


#1

Hi,

I’m finishing a revision of a paper that needs to go to the editor on Monday. One of the reviewers wrote

In Section 3.5.2, line 222, the authors use the expression “finite element cell”, which I think is confusing to the readers, because “cell” belongs to finite volume method. Unless, the author is using a hybrid finite-element finite-volume method, which in that case, the authors should elaborate more on that.

I have written a rather lengthy response, which appears below.

Since my response is long, I’ll ask my questions first. Can someone, perhaps one of the authors of the GJI papers [3] and [5] below, comment on what I’ve written. Also,

  1. Are there references in the general FEM literature in which “cell” and “element” are used in the same manner as in the ASPECT literature and, if so, would you please give me a pointer to two or three of them?

  2. Is what I have written in italics towards the end (before I reference [2] Brenner and Scott) correct or even partially correct?

  3. Should I simply end my response after my first paragraph; the one that has as part of it’s last sentence:

i.e., we want to remain consistent with the literature that the community of ASPECT users, or potential users, typically read.

  1. Suggestions? I just want the reviewer to accept the paper as soon as they see it. I don’t want to insult them or offend them.

The reason we use the word “cell” is that in the peer reviewed articles that describe ASPECT in detail [3, 5] as well as in the ASPECT manual [1] the word “cell” is used to describe a single component of the computational grid; i.e., a cell. In particular, at the time and date this is being written, the version of the

ASPECT Manual

which is updated once a day as part of the continual development of ASPECT has 313 occurrences of the word “cell”. This is our principal reason for referring to these objects (finite element “cells”) as “cells”; i.e., we want to remain consistent with the literature that the community of ASPECT users, or potential users, typically read.

We are not sure what word the reviewer would like us to use instead of “cell”. We
have found instances of lecture notes on the web; for example,

M1ApproxPDE_Chapter5-1.pdf

where the author uses the word “element” to refer to what we call a cell; e.g., in the second sentence on page 124. We have little doubt that the word“element” is given this meaning in both books and peer reviewed articles on the FEM.

Nevertheless, in the discussion in Section 3.2.2 in the peer reviewed journal article [5] that describes the initial implementation of ASPECT in detail and, more specifically, starting at the bottom of page 16 and in the footnote numbered ‘2’ at the bottom of page 17 the word “element” is used in the following manner.

“ 2 Note, however, that the eight pressure nodes per cell in 3-D for the Q_1 element are shared between all cells adjacent to each vertex, whereas the three pressure nodes per cell for the P_{−1} element are uniquely associated
with each cell. …” [all italics are mine]

Similar uses of the word ‘element’ may be found in the second, peer reviewed article [3] from 2017 describing the current version of ASPECT, including changes from the original version, additional capabilities, etc.

As an aside, we note that there may be no universally accepted definition of what the word “element” in Finite Element Method means.

For example, in the mathematics community, in which many of us work, the definition of element that is due to Professor Philippe G. Ciarlet, is (loosely) as follows; e.g., see [2].
The formal definition is presented as a triple:
i. K: A “nice enough” closed subset of R^n ,
ii. P: A finite dimensional function space on K,
iii. N: “nodes”, a basis for the dual space P^∗
The “nodal basis” is then defined as the function φ ∈ P such that n_i (φ j ) = δ_ij. Nodes are often pointwise evaluation; i.e., n_i (φ) = φ(x_i ) for some point x_i ∈ K, but much more general n_i are possible.

[1] Bangerth, W., Dannberg, J., Gassmöller, R., Heister, T., and et al. ASPECT: Advanced Solver for Problems in Earth’s ConvecTion User Manual. Computational Infrastructure for Geodynamics, 2018.
[2] Brenner, S. C., and Scott, R. The Mathematical Theory of Finite Element Methods, Third ed. No. 15 in Texts in Applied Mathematics. Springer-Verlag, New York, 2008.
[3] Heister, T., Dannberg, J., Gassmöller, R., and Bangerth, W. High accuracy mantle convection simulation through modern numerical methods. II: Realistic models and problems. Geophys. J. Int. 210, 2 (2017), 833–851.
[5] Kronbichler, M., Timo Heister, and Wolfgang Bangerth. High accuracy mantle convection simulation through modern numerical methods. Geophys. J. Int. 191, 1 (2012), 12–29.


#2

Gerry,
historically, the word “element” in “finite element method” was used to
indicate that the method works by decomposing a solid (which is where the FEM
was originally developed) into a set of finitely-sized elements in much the
same way as you would decompose a truss bridge into its individual beams
(elements). It was meant as an analogy: you break a body into a finite number
of simpler mechanical elements.

This meaning was lost as the FEM was used more widely and the term “cell” is
now pretty much universally lost as an item in the mesh – more or less
equivalent to the old meaning of the word “element”. So I think the reviewer’s
comment that the term is unclear is not correct: everyone using the finite
element method will know what a “cell” is, and the term cell is most
definitely not reserved to the finite volume community. I do think that the
term “finite element cell” is poorly chosen – the last word is in essence
duplicated. I would just say “cell”, or if you want “a cell in the finite
element mesh” or “a cell in the triangulation”.

Best
W.


#3

Thank you! How would you respond to the reviewer? The first paragraph seems adequate to me, but it doesn’t make the point you made in your reply. I could also quote your answer above. In particular,

… everyone using the finite element method will know what a “cell” is, and the term cell is most definitely not reserved to the finite volume community.

However, I am concerned I may offend the reviewer.

I don’t know if you looked at the link

M1ApproxPDE_Chapter5-1.pdf

but that author clearly uses the word element where we would use ‘cell.’ But this is probably a tangent. However this is why I concluded

As an aside, we note that there may be no universally accepted definition of what the word “element” in Finite Element Method means.

Do you agree?


#4

Thank you! How would you respond to the reviewer? The first paragraph seems
adequate to me, but it doesn’t make the point you made in your reply. I could
also quote your answer above. In particular,

… everyone using the finite element method will know what a “cell” is, and
the term cell is most definitely not reserved to the finite volume community.

However, I am concerned I may offend the reviewer.

Then write “I suspect that everyone using…”, or “From my experience working
in this community, I am fairly convinced that everyone using…”, or “I have
asked around, and I believe that everyone using…” – it’s all just a
question of wording the same thing in different layers of obfuscation :wink:

M1ApproxPDE_Chapter5-1.pdf
https://www.ljll.math.upmc.fr/ledret/M1English/M1ApproxPDE_Chapter5-1.pdf

but that author clearly uses the word element where we would use ‘cell.’ But
this is probably a tangent. However this is why I concluded

/As an aside, we note that there may be no universally accepted definition of
what the word “element” in Finite Element Method means./

Do you agree?

No. The words “element” and “cell” are used interchangeably. I believe that
the definition of both words is identical, and that that definition is
universally accepted. Both words mean the same thing, and everyone agrees.

What people don’t agree on is how that concept is called – some use the word
“element” and others use the word “cell”. It’s like people calling that thing
on four wheels that drives down the road either “car” or “automobile”. Both
mean the same, and neither word is ambiguous.

Best
W.